"I make my loops small".....
Like several mentioned, a sharp (or less rounded) edge across the groove path, along which the string loop follows, is very common. But, also, a smaller string loop, naturally, lays the string more acutely around any sharp edge. A more acute string angle (due to the smaller diameter loop) over any edge increases force between the edge and string. Greater force over a smaller area (edge) = increased pressure. Increased pressure cuts strings. A larger loop would decrease pressure, due to its obtuse angle, but the real problem is usually a sharp edge perpendicular to the string, how/where yours is cut.
I like smaller loops, too, but without a sharp edge, small loops are not a problem. Scissors cut strings really well, when that string is bent around its two edges.